In a multivariate normal model, what are the consequences of having a large marginal variance (diagonal element) for each random variable? Suppose that we have that $X_1, \ldots, X_n \sim N(\mathbf{0}, \mathbf{\Sigma})$ where $\mathbf{0}$ is an $n \times 1$ vector and $\mathbf{\Sigma}$ is a covariance matrix. Suppose that we let the diagonal elements of $\mathbf{\Sigma}$ be large, like $25$, while keeping the rest of the off-diagonals some correlation parameter $\rho$. I am wondering what the difference is between having a large diagonal element in $\mathbf{\Sigma}$ vs having a smaller diagonal element, like $1$ in the diagonals instead. 
It seems asymptotically it might lead to slower convergence when calculating the MLE and associated estimators. However, are there any other reasons why usually people don't have a large variance element and like to keep it at $1$?
 A: If the off-diagonal correlations $\rho$ are all the same number, then you have
$$
\Sigma = \begin{bmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 & \rho\sigma_1\sigma_3 & \cdots \\ \rho\sigma_2\sigma_1 & \sigma_2^2 & \rho\sigma_2\sigma_3 & \cdots \\
\rho\sigma_3\sigma_1 & \rho\sigma_3\sigma_2 & \sigma_3^2 & \cdots \\
\vdots & \vdots & \vdots \end{bmatrix}.
$$
Not suppose you just mutltiply all of the diagonal elements, which are the marginal variances, by $25.$ Then all of the off-diagonal entries also get multiplied by $25,$ and the whole matrix becomes $25\Sigma.$ In effect, if the matrix above is $\operatorname{var} X,$ where $X$ is an $n\times1$ random vector, then $25\Sigma = \operatorname{var} (5X).$ So the whole thing just gets re-scaled.
However, since you say the marginal variances are what you're concerned with, perhaps you wanted to know what happens if you multiply all of those by some large number while leaving the off-diagonal entries as they are. But in that case, the correlation does not remain the same: If one changes $\sigma_i^2$ to $25\sigma_i^2$ for $i=1,\ldots,n,$ while leaving the $(i,j)$ entry as $\rho\sigma_i\sigma_j$ for $i\ne j,$ then the correlation is no longer $\rho,$ but $\rho/25.$ In effect, you're making the correlations approach $0$ as you increase the factor by which you multiply the diagonal elements. The dependence among the components of $X$ gets weaker.
